Theorem bj-2upleq 32193

Description: Substitution property for ⦅ − , − ⦆. (Contributed by BJ, 6-Oct-2018.)

Assertion

Ref	Expression
bj-2upleq	⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Proof of Theorem bj-2upleq

Step	Hyp	Ref	Expression
1		bj-1upleq 32180	. . 3 ⊢ (𝐴 = 𝐵 → ⦅𝐴⦆ = ⦅𝐵⦆)
2		bj-xtageq 32169	. . 3 ⊢ (𝐶 = 𝐷 → ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷))
3		uneq12 3724	. . . 4 ⊢ ((⦅𝐴⦆ = ⦅𝐵⦆ ∧ ({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷)) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
4	3	ex 449	. . 3 ⊢ (⦅𝐴⦆ = ⦅𝐵⦆ → (({1𝑜} × tag 𝐶) = ({1𝑜} × tag 𝐷) → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))))
5	1, 2, 4	syl2im 39	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))))
6		df-bj-2upl 32192	. . 3 ⊢ ⦅𝐴, 𝐶⦆ = (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶))
7		df-bj-2upl 32192	. . 3 ⊢ ⦅𝐵, 𝐷⦆ = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷))
8	6, 7	eqeq12i 2624	. 2 ⊢ (⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆ ↔ (⦅𝐴⦆ ∪ ({1𝑜} × tag 𝐶)) = (⦅𝐵⦆ ∪ ({1𝑜} × tag 𝐷)))
9	5, 8	syl6ibr 241	1 ⊢ (𝐴 = 𝐵 → (𝐶 = 𝐷 → ⦅𝐴, 𝐶⦆ = ⦅𝐵, 𝐷⦆))

Syntax hints:   → wi 4   = wceq 1475   ∪ cun 3538  {csn 4125   × cxp 5036  1_𝑜c1o 7440  tag bj-ctag 32155  ⦅bj-c1upl 32178  ⦅bj-c2uple 32191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-un 3545  df-opab 4644  df-xp 5044  df-bj-sngl 32147  df-bj-tag 32156  df-bj-1upl 32179  df-bj-2upl 32192

This theorem is referenced by:  bj-2uplth  32202